The centroid is the average of all points of a shape. It is therefore the only point that contains information about the entire shape. Any other point doesn't. Naturally, since paths are distances and can also be averaged in the same way, it is actually quite obvious that the path travelled by the centroid of a shape is also the average path of all the points on that shape. In fact, it directly follows from the definition of a centroid consider that the path of a shape is the addition of one more dimensional coordinate to each of the points on that shape.

Sorry,I still do not see what makes this so special.